Partial bosonization for the two-dimensional Hubbard model

Partial bosonization of the two-dimensional Hubbard model focuses the functional renormalization flow on channels in which interactions become strong and local order sets in. We compare the momentum structure of the four-fermion vertex, obtained on the basis of a patching approximation, to an effective bosonic description. For parameters in the antiferromagnetic phase near the onset of local antiferromagnetic order, the interaction of the electrons is indeed well described by the exchange of collective bosonic degrees of freedom. The residual four-fermion vertex after the subtraction of the bosonic-exchange contribution is small. We propose that similar partial bosonization techniques can improve the accuracy of renormalization flow studies also for the case of competing order.

Figure 1

Momentum dependence of the four-fermion vertex. Parameters are $\mu =t^{\prime }=0$, $U/t=3$, $T=0.1$, and the vertex is evaluated at $k_{\text{IR}}\approx 0.9t$ ($T_k\approx 0.25t$). We display $U_k(P,P,-P,-P)=U_{\text{CPH}}\left(P\right)$, where at one-loop level only the crossed particle-hole channel (CPH) contributes, with $P=(0,p_x,p_y)$. The top left panel shows the numerical result, which is dominated by nesting peaks located at $(\pm \pi /2,\pm \pi /2)$. The top right panel shows a boson-exchange mediated vertex of the form ${\lambda }_k^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}=2{(m_k^2+A_k{[2\mathit{p}-\mathit{\pi }]}^2)}^{-1}+{\lambda }_k^{\left(0\right)}$, where $m_k^2$, $A_k$, and ${\lambda }_k^{\left(0\right)}$ are momentum-independent fitting parameters. The bottom panels show the momentum dependence of the residual four-fermion interaction, given by the difference between the two vertices on the top panels, $\mathrm{\Delta }{\lambda }_k^{\left(\psi \right)}\left(\mathit{p}\right)=U_{\text{CPH}}\left(P\right)-[{\lambda }_k^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}]$. The bottom right panel is a magnified version of the one on the bottom left. Differences between $U_{\text{CPH}}\left(P\right)$ and ${\lambda }_k^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}$ are overall about one order of magnitude smaller than the peaks.

Figure 2

Four-fermion coupling $U_k(P,-P,-P,P)=U_{\text{DPH}}\left(P\right)$, $P=(0,p_x,p_y)$, where at one-loop level only the direct particle-hole channel (DPH) contributes. Parameters are the same as for Fig. 1. The numerical result shown in the top left panel is dominated by nesting peaks located at $(\pm \pi /2,\pm \pi /2)$. The top right panel shows a bosonic-exchange propagator of the form ${\lambda }_{\text{DPH}}^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}={(m_k^2+A_k{[2\mathit{p}-\mathit{\pi }]}^2)}^{-1}+{\lambda }_{\text{DPH}}^{\left(0\right)}$ fitted to the result. We find that $U_{\text{DPH}}\left(P\right)\approx {\lambda }_{\text{DPH}}^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}$ to high accuracy. The bottom panels show the the momentum dependence of the residual four-fermion vertex as difference between $U_{\text{DPH}}\left(P\right)$ and ${\lambda }_{\text{DPH}}^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}$, where the bottom right panel is a zoomed-in version of the one on the bottom left. Differences between $U_{\text{DPH}}\left(P\right)$ and ${\lambda }_{\text{DPH}}^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}$ are overall about one order of magnitude smaller than the peaks.

Figure 3

Four-fermion coupling $U_k(P,P,P,P)=U_{\text{PP}}\left(P\right)$, $P=(0,p_x,p_y)$, where at one-loop level only the particle-particle channel (PP) contributes. The numerical result shown in the top left panel is qualitatively different from the one in the particle-hole channels. Instead of nesting peaks, we observe a suppression of $U_k$ at $p_x=0,\pm \pi$ and $p_y=0,\pm \pi$. The top right panel shows a bosonic-exchange propagator of the form ${\lambda }_{\text{PP}}^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}={(m_k^2+A_k{\left[2\mathit{p}\right]}^2)}^{-1}+{\lambda }_{\text{PP}}^{\left(0\right)}$ fitted to the result. We find that $U_{\text{PP}}\left(P\right)\approx {\lambda }_{\text{PP}}^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}$ is a good approximation. The bottom panels show the difference between $U_{\text{PP}}\left(P\right)$ and ${\lambda }_{\text{PP}}^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}$, where the bottom right panel is a zoomed-in version of the one on the bottom left. We find that differences between $U_{\text{PP}}\left(P\right)$ and ${\lambda }_{\text{PP}}^{\left(b\right)}\left(\mathit{p}\right)+U_k^{\left(0\right)}$ are much smaller than the peaks.

Figure 4

Fit parameters $m_k^2$ and $A_k$ for the boson-exchange propagators presented in Sec. 6e at different renormalization scales $k$. In all cases the values decrease exponentially toward the IR, signaling a buildup of nesting peaks, or more generally a nontrivial momentum dependence of the four-fermion coupling which can be described as an effective boson exchange. The parameters for the two particle-hole channels tend toward each other, consistent with the exchange of an antiferromagnetic boson in the two channels.